Statistical Tests & Distribution Calibration¶

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Abstractions about test statistics, null distributions, consistency, randomness tests, t-tests, quantile normalization, and universal hypothesis testing.

7 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

Fisher Consistency — A population-level calibration property requiring an estimator or decision rule, viewed as a functional, to recover the target parameter or Bayes-optimal action when applied to the true data-generating distribution.
Null distribution — Represent the sampling distribution of a declared test statistic under the null hypothesis and sampling scheme used to calibrate tail probabilities, critical values, and type-I error.
Quantile normalization — Replace values by a shared rank-indexed reference so multiple samples have the same empirical marginal distribution while preserving within-sample rank order.
Randomness Test — Challenge a sequence against a specified stochastic null using a pattern-sensitive statistic and calibrated rejection rule, while treating a pass only as failure to detect the tested departures.
Statistic — A measurable function of the observed sample alone, with no dependence on unknown population parameters, used to summarize data or support estimation and testing.
Student's t-Test — A family of mean-inference procedures that divides an observed mean or mean difference by its estimated standard error and evaluates the resulting statistic against a Student t distribution whose degrees of freedom account for estimating variance from the sample.
Universal Hypothesis Testing — A goodness-of-fit testing problem that compares one fully specified null distribution with the unrestricted alternative of every other distribution, seeking level-controlled tests that remain consistent or error-exponent optimal without modeling a particular alternative.